On the properties of functions of the Takagi power class

TL;DR

This paper investigates the mathematical properties of functions within the Takagi power class, focusing on their continuity, differentiability, and extremal points for different parameter values.

Contribution

It provides a detailed analysis of the properties of Takagi power class functions, including conditions for continuity, differentiability, and maximum points, expanding understanding of these fractal-like functions.

Findings

Functions are continuous for all p > 0.

Differentiability depends on the parameter p and specific points.

Global maxima occur at certain points depending on p.

Abstract

By construction, functions of Takagi power class are similar to Takagi's continuous nowhere differentiable function. These functions have one real parameter $p>0$. They are defined by the series $S_p(x) = \sum_{n=0}^\infty (T_0(2^nx)/2^n)^p$, where $T_0(x)$ is the distance from the real number $x$ to the nearest integer. We study such properties of functions $S_p(x)$ as continuity, generalized H\"older condition, global maxima and differentiability at the points $x=1/3$ and $x=2/3$, for various values of $p$.